\(\int \frac {c+d x}{a+b x} \, dx\) [1241]

Optimal result

Integrand size = 13, antiderivative size = 25 \[ \int \frac {c+d x}{a+b x} \, dx=\frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2} \]

[Out]

d*x/b+(-a*d+b*c)*ln(b*x+a)/b^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {c+d x}{a+b x} \, dx=\frac {(b c-a d) \log (a+b x)}{b^2}+\frac {d x}{b} \]

[In]

Int[(c + d*x)/(a + b*x),x]

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx \\ & = \frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x}{a+b x} \, dx=\frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2} \]

[In]

Integrate[(c + d*x)/(a + b*x),x]

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^2

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04

method	result	size
default	\(\frac {d x}{b}+\frac {\left (-a d +b c \right ) \ln \left (b x +a \right )}{b^{2}}\)	\(26\)
norman	\(\frac {d x}{b}-\frac {\left (a d -b c \right ) \ln \left (b x +a \right )}{b^{2}}\)	\(27\)
parallelrisch	\(-\frac {\ln \left (b x +a \right ) a d -\ln \left (b x +a \right ) b c -b d x}{b^{2}}\)	\(31\)
risch	\(\frac {d x}{b}-\frac {\ln \left (b x +a \right ) a d}{b^{2}}+\frac {c \ln \left (b x +a \right )}{b}\)	\(32\)

[In]

int((d*x+c)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

d*x/b+(-a*d+b*c)*ln(b*x+a)/b^2

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x}{a+b x} \, dx=\frac {b d x + {\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \]

[In]

integrate((d*x+c)/(b*x+a),x, algorithm="fricas")

[Out]

(b*d*x + (b*c - a*d)*log(b*x + a))/b^2

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {c+d x}{a+b x} \, dx=\frac {d x}{b} - \frac {\left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{2}} \]

[In]

integrate((d*x+c)/(b*x+a),x)

[Out]

d*x/b - (a*d - b*c)*log(a + b*x)/b**2

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x}{a+b x} \, dx=\frac {d x}{b} + \frac {{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \]

[In]

integrate((d*x+c)/(b*x+a),x, algorithm="maxima")

[Out]

d*x/b + (b*c - a*d)*log(b*x + a)/b^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x}{a+b x} \, dx=\frac {d x}{b} + \frac {{\left (b c - a d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \]

[In]

integrate((d*x+c)/(b*x+a),x, algorithm="giac")

[Out]

d*x/b + (b*c - a*d)*log(abs(b*x + a))/b^2

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x}{a+b x} \, dx=\frac {d\,x}{b}-\frac {\ln \left (a+b\,x\right )\,\left (a\,d-b\,c\right )}{b^2} \]

[In]

int((c + d*x)/(a + b*x),x)

[Out]

(d*x)/b - (log(a + b*x)*(a*d - b*c))/b^2